The power index of a graph

TL;DR

This paper investigates the power index of graphs, determining the minimal group order for embedding graphs into power graphs, and classifies power-critical graphs within specific families.

Contribution

It classifies all power-critical graphs among complete, bipartite, and 1-factor graphs, and provides conditions for the existence of optimal groups.

Findings

Classified all power-critical graphs in the studied families.

Established necessary and sufficient conditions for mma-optimal groups.

Analyzed power indices for complete, bipartite, and 1-factor graphs.

Abstract

The {\em power index} $\Theta(\Gamma)$ of a graph $\Gamma$ is the least order of a group $G$ such that $\Gamma$ can embed into the power graph of $G$. Furthermore, this group $G$ is {\em $\Gamma$-optimal} if $G$ has order $\Theta(\Gamma)$. We say that $\Gamma$ is {\em power-critical} if its order equals to $\Theta(\Gamma)$. This paper focuses on the power indices of complete graphs, complete bipartite graphs and $1$-factors. We classify all power-critical graphs $\Gamma'$ in these three families, and give a necessary and sufficient condition for $\Gamma'$-optimal groups.